Properties

Label 51870.k3
Conductor 51870
Discriminant 14341017600
j-invariant \( \frac{144215816802121}{14341017600} \)
CM no
Rank 2
Torsion Structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 0, -1092, -13104]); // or
magma: E := EllipticCurve("51870n1");
sage: E = EllipticCurve([1, 1, 0, -1092, -13104]) # or
sage: E = EllipticCurve("51870n1")
gp: E = ellinit([1, 1, 0, -1092, -13104]) \\ or
gp: E = ellinit("51870n1")

\( y^2 + x y = x^{3} + x^{2} - 1092 x - 13104 \)

Mordell-Weil group structure

\(\Z^2 \times \Z/{2}\Z\)

Infinite order Mordell-Weil generators and heights

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(-23, 34\right) \)\( \left(-15, 21\right) \)
\(\hat{h}(P)\) ≈  2.364965992012.00006506923

Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

\( \left(-24, 12\right) \)

Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

\( \left(-24, 12\right) \), \( \left(-23, 34\right) \), \( \left(-15, 21\right) \), \( \left(40, 76\right) \), \( \left(57, 309\right) \), \( \left(552, 12684\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 51870 \)  =  \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 19\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(14341017600 \)  =  \(2^{12} \cdot 3^{4} \cdot 5^{2} \cdot 7 \cdot 13 \cdot 19 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( \frac{144215816802121}{14341017600} \)  =  \(2^{-12} \cdot 3^{-4} \cdot 5^{-2} \cdot 7^{-1} \cdot 13^{-1} \cdot 19^{-1} \cdot 229^{6}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
sage: E.rank()
Rank: \(2\)
magma: Regulator(E);
sage: E.regulator()
Regulator: \(3.7932290999\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(0.835725882117\)
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
Tamagawa product: \( 8 \)  = \( 2\cdot2\cdot2\cdot1\cdot1\cdot1 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
Torsion order: \(2\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 51870.2.a.k

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

\( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{7} - q^{8} + q^{9} - q^{10} - 4q^{11} - q^{12} - q^{13} + q^{14} - q^{15} + q^{16} - 2q^{17} - q^{18} - q^{19} + O(q^{20}) \)

For more coefficients, see the Downloads section to the right.

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 49152
\( \Gamma_0(N) \)-optimal: yes
Manin constant: 1

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
sage: r = E.rank();
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar = ellanalyticrank(E);
gp: ar[2]/factorial(ar[1])

\( L^{(2)}(E,1)/2! \) ≈ \( 6.34019947117 \)

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)[5]
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(2\) \( I_{12} \) Non-split multiplicative 1 1 12 12
\(3\) \(2\) \( I_{4} \) Non-split multiplicative 1 1 4 4
\(5\) \(2\) \( I_{2} \) Split multiplicative -1 1 2 2
\(7\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1
\(13\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1
\(19\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X13.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 6.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime Image of Galois representation
\(2\) B

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

Note: \(p\)-adic regulator data only exists for primes \(p\ge5\) of good ordinary reduction.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type nonsplit nonsplit split nonsplit ordinary nonsplit ordinary nonsplit ss ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 4 6 3 2 2 2 2 2 2,2 2 2 2 2 2 2
$\mu$-invariant(s) 0 0 0 0 0 0 0 0 0,0 0 0 0 0 0 0

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 51870.k consists of 4 curves linked by isogenies of degrees dividing 4.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 \(\Q(\sqrt{1729}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
\(\Q(\sqrt{-91}) \) \(\Z/4\Z\) Not in database
\(\Q(\sqrt{-19}) \) \(\Z/4\Z\) Not in database
4 \(\Q(\sqrt{-19}, \sqrt{-91})\) \(\Z/2\Z \times \Z/4\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.